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Vol. 5, Issue 2, February 2016
Fixed Points of Generalized (
Α, Ψ
)
−
Contractive Maps
Ch.PrabhakaraRao1,. B.CH.K.Preethi2, S.V.V.Rama Devi3, Ch.Pragathi4
Professor Dept. of Mathematics, Dadi Institute of Engineering and Technology, Anakapalle,,Visakhapatnam, A.P, India1
Assistant Professor, Dept. of Mathematics, Dadi Institute of Engineering and Technology, Anakapalle,,Visakhapatnam, A.P,
India 2,3
Associate Professor, Dept. of Mathematics, GITAM Institute of Technology, GITAM University, Visakhapatnam,
A.P, India 4
ABSTRACT: In this paper, we introduce generalized (α,ψ)− contractive maps and prove fixed point theorems in complete metric spaces. Examples are given to illustrate our results. Our results generalize the results of Samet, Vetro, and Verto [1].
KEYWORDS: fixed point, complete metric space, α−admissible, (α,ψ)−Contractive maps.
I. INTRODUCTION
In 2012, Samet, Vetro and Vetro [1] introduced a new concept on contractive maps namely (α,
ψ)−contractive mappings and established fixed point theorems in complete metric space. Here we note
that (α, ψ)-contractive maps generalize contraction maps so that the fixed point results of (α, ψ)- contractive maps generalize Banach contraction principle.
In section 2, we introduce generalized (α, ψ)-contractive maps and prove fixed point theorems in complete metric spaces. Supporting examples are given. Our results generalize the results of Samet, Vetro, Vetro, [1] and Banach contraction principle.
Throughout this paper we denote
Ψ= { : [0, +∞)→[0, +∞)| ,∑∞ ( ) <∞∀ > 0 }.
Lemma 1.1. Let ψ ∈Ψ. Then ψ (t) < t for all t > 0. Remark 1.2. → ( ) = .
Definition 1.3. [1] Let (X, d) be a metric space. Let T : X → Xbe a self map of X.If there exist two functions α : X × X → [0, ∞) and ψ∈Ψ such that
α(x, y)d(Tx, Ty) ≤ ψ (d(x, y))for all x, y∈X, (1) then T is an (α, ψ)-contractive map.
Definition 1.4. [1] Let T : X → X and α : X×X → [0, ∞). We say that T is α-admissible if α(x, y) ≥ 1 ⇒ α
(Tx, Ty) ≥1 (2)
for all x, y∈X.
The following are the main theorems of Samet, Vetro and Vetro [1]
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contractive map satisfying the following conditions:(i) T is α−admissible, (ii) there exists x0∈X such that
α(x0,Tx0)≥1,and (iii)T is continuous. Then T has a fixed point in X.
Theorem1.6. [1] Let (X, d) be a complete metric space and T : X → X be an
(α, ψ)− contractive map satisfying the following conditions : (i) T is α− admissible, (ii) there exists x0 ∈ X such that α(x0, Tx0) ≥ 1,and (iii) if {xn} is a sequence in X such that α(xn, xn+1) ≥ 1 for all n, and xn → x for all x
∈ X as n → + ∞ then α(xn ,x) ≥ 1 for all n. Then T has a fixed point in X.
To get the uniqueness of fixed point the authors assumed the following hypothesis: (H): for all x, y ∈ X
there exists z ∈ X such that α(x, z) ≥ 1 and α(y, z) ≥1.
Theorem 1.7. [1] Adding condition (H) to the hypotheses of Theorem 1.5 (Theorem 1.6) we obtain the uniqueness of the fixed point of T.
II. FIXED POINTS OF GENERALIZED (Α, Ψ)- CONTRACTIVE MAPS
Definition 2.1. Let (X, d) be a metric space. Let T: X→ Xbe a self map of X. If there exist two functions α: X × X→[0, ∞) and ψ∈Ψ such that
α (x,y) d(Tx,Ty)≤ (max{d(x,y),d(x,Tx),d(y,Ty), ( , ) ( , )}) (3) for all x,y ∈ X, then we say that T is a generalized(α,ψ)-contractive map.
Remark 2.2. Here we note that “every (α, )-contractive map” is a “generalized (α, )−contractive map”. But its converse need not be true.(Example 2.5)
Theorem 2.3. Let (X, d) be a complete metric space and T : X → X be a generalized (α,ψ)−
contractive map satisfying the following conditions : (i) T is α− admissible,
ii) there exists x0∈ X such that α(x0, Tx0) ≥ 1, and (iii) T is continuous. Then T has a fixed point in X that is, there exists x* ∈ X such that Tx* = x*.
Proof. Let x0 ∈ X be such that α(x0, Tx0) ≥ 1 by (ii).We define a sequence {xn} ⊆ X by xn+1 = Txn,n = 0,1,2... . If xn = xn+1 for some n then,
xn = xn+1 = Txn, and xn is a fixed point of T. Hence w.l.g. we suppose that xn ≠xn+1 for all n. Since T is α−admissible, α(x0, x1) = α(x0, Tx0) ≥ 1 ⇒α (Tx0, Tx1) ≥1 so that α(x1, x2) ≥ 1. Now, by induction it is easy to see that α(xn, xn+1) ≥ 1 for all n. (4) By taking x= xn-1and y =xnin(4),we get d(xn, xn+1) = d(Txn-1,Txn) ≤α(xn-1, xn),d(Txn-1,Txn)
≤ (max{d(xn-1,xn),d(xn-1,Txn-1) d(xn,Txn),
( , ) ( , )
})
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By remark 1.2, it follows that, d(xn, xn+1) ≤ ψ(d(xn, xn-1)).It is true for each
n= 1, 2, 3, ... and hence it follows that d(xn, xn+1) ≤ ψn (d(x1, x0)) for each n = 1, 2, 3…
Let є >0 be given. Since ∑∞ , <∞.
We have lim →∞∑ ( ( , ))=0
Hence there exists N(є) ∈ such that ∑∞ ( , ) <є for all n ≥ N(є).
We now show that {xn} is a Cauchy sequence in X.
Let m, n ∈ Z+ with m > n ≥ N.
d(xn, xm) ≤ d(xn, xn+1) + d(xn+1, xn+2) + .... + d(xm-1, xm)
≤ ψn (d(x1, x0)) + ψ n+1
(d(x1, x0)) + .... + ψ m-1
(d(x1, x0)) = ∑ ( ( , )
≤ ∑ ( ( , ) <
Hence { xn} is a Cauchy sequence in X. Since X is complete, there exists x*∈ X such that lim →∞ = x*.
By (iii) we have T is continuous. x* = lim →∞ = lim →∞ ( ) = T( lim →∞ ) = T x
* . Therefore x* = T x*. Hence the theorem follows.
Remark 2.4 . Theorem 1.5 follows as a corollary to Theorem 2.3, since every (α, ψ ) – contractive map is a’ generalized (α, ψ ) – contractive map’.
Example 2.5. Let X = ℝ with the usual metric. We define T : X → X by
Tx =
⎩ ⎪ ⎪ ⎨ ⎪ ⎪
⎧1, < 0 1− , 0≤ ≤
, < ≤
+ , < ≤1
2 − , > 1
We define α : X × X → [ 0, ∞) by α (x, y) = 1, , ∈[0, ]
0, ℎ
Also, we define ψ : ℝ → ℝ by ψ (t) = , t ≥ 0 . Clearly ψ ∈Ψ . We first show that T satisfies condition (i) – (iii) of Theorem 2.3.
(i) T is α – admissible: for any x,y ∈ X with x,y ∈ [ 0, ] we have α (x, y) ≥ 1.
Hence α ( Tx, Ty ) = α (1-x, ) = 1. So that T is α – admissible.
(ii) there exists x0 ∈ X such that α ( x0 ,Tx0 ) ≥ 1. We choose x0 = , then
α ( , ) = α ( , ) = 1.
(iii) T is continuous on ℝ .We now show that T is generalized ( α, ψ)- contractive map. We have the following four cases.
Case (i) : x ,y ∈ [ 0, ] , then T x = 1-x and T y = 1-y. We assume w.l.g that x ≥ y.
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≤
= ψ ( d(y, Ty ))
≤ ψ ( max { d(x, y),d(x, Tx),d(y, Ty), ( , ) ( , )}) Case (ii) : x ∈ [ 0, ] , y ∈ [ , ] , then T x = 1-x an T y = .
α (x, y ) d(Tx, Ty) = - x
≤ = ψ (d(x, Tx))
≤ ψ ( max { d(x, y),d(x, Tx),d(y, Ty), ( , ) ( , )}) Case (iii) : y ∈ [ 0, ] , x ∈ [ , ] , since the inequality is symmetric in x and y the inequality holds by case (ii).
Case (iv) : x ∈ [ 0, ] , y ∈ [ , 1] , then T x = 1- x, Ty = + . This case is trivial, since α (x, y) = 0.
Therefore T is generalized ( α, ψ)- contractive map. Thus T satisfies all the hypotheses of Theorem 2.3 and T has two fixed points . Here we obtain
that when x = 0 and y = we have T x = 1 and T y = , and hence
α (x, y) d(T x, Ty) = α (0, ) d(1, ) = ≮ ψ (d (0, )) = ψ (d(x, y)) for any ψ∈Ψ. Hence T is not an (α, ψ) – contractive map, so that theorem 1.5 is not applicable. Hence Theorem 2.3 generalizes Theorem 1.5.
Theorem 2.6. Let (X, d) be a complete metric space and T : X → X be a generalized (α,ψ)− contractive map satisfying the following conditions: (i) T is α−
admissible, (ii) there exists x0∈X such that α(x0,T x0)≥ 1, and (iii) if { x n } is a sequence in X such that α (xn, xn+1) ≥ 1 for all n, and xn → x ∈X as n → +∞ then α (xn, x) ≥ 1 for all n. (iv) T is continuous, Then T has a fixed point in X.
Proof. On proceeding as in the proof of Theorem2.3, we get { xn } is a Cauchy sequence in X. Since X is complete, there exists x* ∈ X such lim →∞ = x*. In
the proof of Theorem 2.3, we proved that (xn ,xn+1) ≥1 for all n. Hence from (iii) we have (xn , x* ) ≥ 1 for all n. Suppose that T x* ≠ x* . We consider
d(T x*, x* ) ≤ d(T x*, T xn ) + d(T xn , x* )
≤ α ( x8 , xn )d (T x* ,T xn) + d(T xn , x*) ≤ ψ (max{d( x* , xn),d( x*,T x*),d(xn,Txn),
( ∗, ) ( , ∗)
}) + d(xn+1, ∗)
On letting n →∞, we get d( T x*, x*) ≤ ψ(max{0,d( x*,T x*),0, ( ∗, ∗)
})+0
= ψ(d( x*,T x* ) < d( x*,T x*),a contradiction.
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Example 2.7. Let X = ℝ with the usual metric. We define T : X → X by
Tx =
⎩ ⎪ ⎪ ⎨ ⎪ ⎪
⎧1, < 0 1− , 0≤ ≤
, < ≤
1 , < ≤1
2 − , > 1
We define α : X × X → [ 0, ∞) by α (x, y) = 1, , ∈[0, ]
0, ℎ
Also, we define ψ : ℝ → ℝ by ψ (t) = , t ≥ 0 . Clearly ψ ∈ Ψ .Clearly T is not continuous on ℝ . As
in example 2.5, it is easy to see that T is a generalized (α, ψ)- contractive map. Clearly T is α – admissible and for x0 = , we have
α (x0 ,Tx0) = α ( , T ) = α ( , ) = 1. Hence conditions (i) and (ii) of Theorem 2.6 hold. Let
{ xn } ⊆ X such that α (xn , xn+1) ≥ 1. Then by the definition of α, we have xn ∈ [0, ] for all n.
Now if xn → x as n → ∞, since [0, ] is closed, it follows that x ∈ [0, ] and hence α(xn , x ) ≥ 1 for all n. Thus condition (iii) of Theorem 2.6 holds.
Thus T satisfies all the hypotheses of Theorem 2.6 and T has three fixed points 1, , and .
We now prove the uniqueness of fixed points of generalized ( α, ψ )- contractive maps, by assuming the hypotheses(H).
Theorem2.8. Adding the hypotheses (H) to the hypotheses of Theorem 2.3 (Theorem2.6), we obtain the uniqueness of the fixed point of T.
Proof. By Theorem 2.3 ( Theorem2.6 ), T has a fixed point in X. Suppose that T has two fixed points x* , y* (say)
in X. By above condition (H), there exists z in X such that α (x*, z) ≥ 1 and α (y*, z) ≥ 1. (5)
Since T is α – admissible and from (5) we get α (x*, Tn z) ≥ 1 and α(y*, Tn z) ≥ 1 (6)
Now by generalized (α, ψ )- contractive map and using (6), we get d(x*,Tn z) = d(Tx*, T(Tn-1z)) ≤ α (x*,Tn-1z) d(Tx* ,T(Tn-1z))
≤ ψ (max {d(x*,Tn-1z),d(x* Tx* ),d(Tn-1z,Tnz), (
∗, ) ( , ∗) })
≤ ψ (max {d(x*,Tn-1z), 0 ,d(Tn-1z,Tnz), (
∗, ) ( , ∗) })
≤ ψ (max {d(x*,Tn-1z), 0 ,d(Tn-1z, x* ), d(x*, Tnz ) }) ≤ ψ ( d(x*,Tn-1z) )
. .
.
≤ ψn (d(x*, z) ) for all n . Thus on letting n → ∞ , we get Tn z → x* . By a similar argument we can show that Tnz → y*
.Therefore by the uniqueness of the limits , we have x* = y* . Therefore T has a unique fixed point in X.
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Tx =
⎩ ⎪ ⎪ ⎨ ⎪ ⎪
⎧1, < 0 1− , 0≤ ≤
, < ≤
+ , < ≤1
− , > 1
We define α : X × X → [ 0, ∞) by α (x, y) = 1, , ∈[0, ]
0, ℎ
Also, we define ψ : ℝ → ℝ by ψ (t) = , t ≥ 0 . Clearly ψ ∈Ψ. As in example 2.5, it is easy to see that T is a
generalized (α, ψ ) contractive map. Clearly T is α – admissible and for x0 = , we have α (x0 ,Tx0) = α (, T ) = α ( , ) = 1. Clearly T is continuous on ℝ and condition (H) holds trivially. Hence T satisfies all the hypotheses of
Theorem 2.8 and T has a unique fixed point .
REFERENCES
